Question

in the given figure DEFG is a square and angle BAC =90°.show that DE^2 =BD×ED​

Answer 1

FIGURE IS IN THE ATTACHMENT.

Given: DEFG is a square and ∠BAC = 90°.

To Prove: DE² = BD × EC.

Proof :

In ∆ AFG & ∆DBG

∠GAF = ∠BDG     [ 90°]

∠AGF = ∠DBG     [corresponding angles because GF|| BC and AB is the transversal]

∆AFG ~ ∆DBG [by AA Similarity Criterion] …………(1)

In ∆ AGF & ∆EFC

∠AFG = ∠CEF     [ 90°]

∠AFG = ∠ECF    [corresponding angles because GF|| BC and AC is the transversal]

∆AGF ~ ∆EFC [by AA Similarity Criterion] …………(2)

From equation 1 and 2.

∆DBG ~ ∆EFC

BD/EF = DG /EC

BD/DE = DE /EC    [ DEFG is a square]

DE² = BD × EC .

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